Fractal structure of the Fibonacci Ising quasicrystal: uniform multifractality of Lee-Yang zeros and single-fermion spectrum at arbitrary couplings, and the topology of finite-fermion spectra below the interval threshold

For the one-dimensional Ising model whose couplings are modulated by the Fibonacci substitution (the Fibonacci Ising quasicrystal), the zeros of the partition function in the complex fugacity plane (Lee-Yang zeros) and the single-fermion spectrum of the associated transverse-field quantum chain both converge, in the thermodynamic limit, to Cantor sets on the unit circle and the real line respectively, and both are governed by the hyperbolic dynamics of the Fibonacci trace map. The rigorous theory (Yessen 2014) established that these limit sets are uniform multifractal sets (um-sets), but the stronger statement needed for a complete fractal characterization — that they are moderate um-sets, i.e. that the box-counting dimension exists and coincides with the Hausdorff dimension — was proven only when the two coupling values are sufficiently close to each other. It was conjectured that this restriction is unnecessary. Two further open questions from the same work — independence of the limiting Lee-Yang zero set of the choice of sequence in the substitution hull, and the absence of isolated points in that set — were subsequently resolved by Damanik, Munger and Yessen through a correspondence between Ising transfer matrices and CMV matrices (the unitary analogs of Jacobi matrices from orthogonal polynomials on the unit circle). The surviving open core, stated here, is: (i) prove or disprove that the thermodynamic-limit Lee-Yang zero set and the quantum single-fermion spectrum are moderate um-sets for all positive distinct couplings, a positive answer being expected to follow from transversality of the curve of initial conditions of the Fibonacci trace map with the center-stable manifolds at every admissible parameter; and (ii) determine the topological structure of the N-fermion spectra below the threshold beyond which they become intervals, where a Cantorval (a compact set with dense interior, informally a 'Cantor set with dense interior') has been conjectured to occur.

Background

Take a binary alphabet $\{a,b\}$ with the Fibonacci substitution $S(a)=ab$, $S(b)=a$, and let $\Omega$ be the hull of the substitution: the compact set of all two-sided sequences obtained as limits of iterates of $S$, equipped with the left shift. Each sequence $\omega=(\omega_i)\in\Omega$ labels lattice sites by letters and thus assigns to each site one of two positive coupling values $p(a),p(b)$ with $p(a)\ne p(b)$; the resulting aperiodic spin chain is the Fibonacci Ising quasicrystal. For the classical chain of $N$ spins $\sigma_i\in\{\pm1\}$ with Hamiltonian $H(\sigma)=-\sum_{i=1}^{N-1}J_i\sigma_i\sigma_{i+1}-h\sum_i\sigma_i$, the Lee-Yang theorem (Lee and Yang, 1952) states that for positive (ferromagnetic) couplings all zeros of the partition function $Z_N$, viewed as a function of the fugacity $\eta=e^{-2\beta h}$, lie on the unit circle $S^1$; the limiting configuration of these zeros as $N\to\infty$ controls the analytic structure of the free energy and hence possible phase transitions. Baake, Grimm and Pisani (1995) and Barata and Goldbaum (2001) studied these zeros for aperiodic couplings numerically and heuristically, observing Cantor-like limit sets with nontrivial gap structure, but rigorous fractal statements were missing until Yessen (2014).

Two parallel objects arise. In the quantum transverse-field Ising chain with Fibonacci-modulated couplings, the Jordan-Wigner transformation maps the model to free fermions, and the single-fermion excitation spectrum $B_\infty$ is the spectrum of a tridiagonal (Jacobi-type) Fibonacci Hamiltonian. In the classical model, the transfer-matrix formalism connects to orthogonal polynomials on the unit circle (OPUC): CMV matrices are the canonical five-diagonal unitary matrices generated by Szegő's recursion, parametrized by Verblunsky coefficients $\alpha_n$ in the unit disk, and Damanik, Munger and Yessen (2013) established a constructive correspondence identifying the thermodynamic-limit Lee-Yang zero set with the essential spectrum of a Fibonacci CMV matrix. Both problems are controlled by the hyperbolic dynamics of the Fibonacci trace map $T(x,y,z)=(2xy-z,x,y)$ on its invariant surfaces $S_V=\{x^2+y^2+z^2-2xyz=V\}$; the key mechanism upgrading qualitative Cantor structure to quantitative fractal statements is transversality of the curve of initial conditions of the model with the center-stable manifolds of $T$.

A Cantor set here means a nonempty compact set with no isolated points whose complement is dense. Following Yessen, a Cantor set $S$ is a um-set (uniform multifractal set) if it has zero Lebesgue measure, its Hausdorff dimension lies strictly between $0$ and $1$, and the local Hausdorff dimension $s\mapsto\dim_H^{\mathrm{loc}}(S,s)$ is continuous and non-constant on every neighborhood within $S$ — the set is a multifractal uniformly at every point. A um-set is moderate if in addition its box-counting dimension exists and coincides with its Hausdorff dimension; this is automatic for dynamically defined Cantor sets but not in general. Yessen proved that $B_\infty$ and the Lee-Yang zero limit set $\mathcal Z\subset S^1$ are um-sets for every choice of positive distinct couplings, that they are moderate um-sets only under a small-parameter restriction (with $p(a)$ fixed, $p(b)$ sufficiently close to $p(a)$), and stated the belief that the restriction can be dropped. For $N$ noninteracting fermions the energy spectrum $B_\infty^N$ is the closure of the set of sums of $N$ distinct elements of $B_\infty$; a theorem of Yessen gives, for every positive distinct coupling pair, a threshold $N_0$ such that $B_\infty^N$ is a closed interval for all $N\ge N_0$ (with $N_0=2$ available near $p(a)=p(b)$), leaving the topology of the sub-threshold spectra $N<N_0$ undetermined. A Cantorval is a compact set with dense interior such that any connected component of its complement has endpoints that are approximated by endpoints of other such components — informally, a 'Cantor set with dense interior'; such sets arise naturally in dynamical systems and number theory.

Problem Statement

Consider the one-dimensional Fibonacci Ising quasicrystal: the Ising spin chain (classical, in a complex magnetic field, and quantum with a transverse field) whose couplings take two positive distinct values $p(a),p(b)$ assigned according to sequences in the hull $\Omega$ of the Fibonacci substitution. Let $\mathcal Z\subset S^1$ denote the thermodynamic-limit set of Lee-Yang zeros of the classical partition function (now known to be independent of the choice of sequence in $\Omega$ and to coincide with the essential spectrum of the associated Fibonacci CMV matrix), and let $B_\infty$ denote the single-fermion spectrum of the quantum chain, i.e. the spectrum of the corresponding tridiagonal Fibonacci Hamiltonian. Two linked questions remain open.

(1) Uniform multifractal structure at arbitrary couplings. It is known that $\mathcal Z$ and $B_\infty$ are um-sets (uniform multifractal Cantor sets) for every positive distinct choice of $p(a),p(b)$, and that they are moderate um-sets — i.e. their box-counting dimension exists and equals their Hausdorff dimension — only under the restriction that one coupling be sufficiently close to the other. Prove or disprove that the closeness restriction can be dropped: for all positive $p(a)\ne p(b)$, are $\mathcal Z$ and $B_\infty$ moderate um-sets? A positive answer is expected to follow from the transversality conjecture for the Fibonacci trace map: that the curve of initial conditions associated with the model intersects the center-stable manifolds of the trace map transversally for all admissible coupling parameters, together with control of the finitely many exceptional points excluded in the known local dimension statements; a negative answer would exhibit, for some coupling pair, a rigorous failure of equality between box-counting and Hausdorff dimensions (or of existence of the former).

(2) Topology of finite-fermion spectra below the interval threshold. For $N\ge N_0$ the $N$-fermion spectrum $B_\infty^N$ (the closure of the set of sums of $N$ distinct elements of $B_\infty$) is a closed interval. Determine the topological structure of $B_\infty^N$ for $2\le N<N_0$ (for coupling pairs for which this range is nonempty), for general positive distinct couplings. In particular, prove or disprove the conjecture that when $p(a)$ and $p(b)$ are sufficiently far apart, $B_\infty^N$ is a Cantorval — a compact set with dense interior whose complement's connected components have endpoints approximated by endpoints of other such components — for some $2\le N<N_0$. Question (2) asks for the topology of $B_\infty^N$ over the full range $2\le N<N_0$ and for all positive distinct couplings; settling the far-apart Cantorval case for a single $N$ is a component of, and not by itself a complete resolution of, that determination.

Both parts serve one objective: the complete fractal and topological characterization of the spectral sets of the Fibonacci Ising quasicrystal over the entire space of positive distinct couplings, removing the remaining small-parameter restrictions.

References

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